An abstract approach to some spectral problems of direct sum differential operators
Date
2003-07-10
Authors
Sokolov, Maksim S.
Journal Title
Journal ISSN
Volume Title
Publisher
Southwest Texas State University, Department of Mathematics
Abstract
In this paper, we study the common spectral properties of abstract self-adjoint direct sum operators, considered in a direct sum Hilbert space. Applications of such operators arise in the modelling of processes of multi-particle quantum mechanics, quantum field theory and, specifically, in multi-interval boundary problems of differential equations. We show that a direct sum operator does not depend in a straightforward manner on the separate operators involved. That is, on having a set of self-adjoint operators giving a direct sum operator, we show how the spectral representation for this operator depends on the spectral representations for the individual operators (the coordinate operators) involved in forming this sum operator. In particular it is shown that this problem is not immediately solved by taking a direct sum of the spectral properties of the coordinate operators. Primarily, these results are to be applied to operators generated by a multi-interval quasi-differential system studied, in the earlier works of Ashurov, Everitt, Gesztezy, Kirsch, Markus and Zettl. The abstract approach in this paper indicates the need for further development of spectral theory for direct sum differential operators.
Description
Keywords
Direct sum operators, Cyclic vector, Spectral representation, Unitary transformation
Citation
Sokolov, M. S. (2003). An abstract approach to some spectral problems of direct sum differential operators. <i>Electronic Journal of Differential Equations, 2003</i>(75), pp. 1-6.
Rights
Attribution 4.0 International